Tauberian conditions for w-almost convergent double sequences

Tauberian conditions for w-almost convergent double sequences
Kuo, Meng-Kuang
2009-02-06 00:00:00
In this paper, we introduce the concept of w-almost convergent sequences. Such a definition is a weak form of almost convergent sequences given by G. G. Lorentz in [Acta Math. 80(1948),167-190]. We give a detailed study on w-almost convergent double sequences and prove that w-almost convergence and almost convergence are equivalent under the boundedness of the given sequence. The Tauberian results for w-almost convergence are established. Our Tauberian results generalize a result of Lorentz and Tauber’s second theorem. Moreover, we prove that w-almost convergence and norm convergence are equivalent for the sequence of the rectangular partial sums of the Fourier series of f ∈ L
p
(T
2), where 1 < p < ∞.
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngPositivitySpringer Journalshttp://www.deepdyve.com/lp/springer-journals/tauberian-conditions-for-w-almost-convergent-double-sequences-qsAQqhDUl1

Abstract

In this paper, we introduce the concept of w-almost convergent sequences. Such a definition is a weak form of almost convergent sequences given by G. G. Lorentz in [Acta Math. 80(1948),167-190]. We give a detailed study on w-almost convergent double sequences and prove that w-almost convergence and almost convergence are equivalent under the boundedness of the given sequence. The Tauberian results for w-almost convergence are established. Our Tauberian results generalize a result of Lorentz and Tauber’s second theorem. Moreover, we prove that w-almost convergence and norm convergence are equivalent for the sequence of the rectangular partial sums of the Fourier series of f ∈ L
p
(T
2), where 1 < p < ∞.

Journal

Positivity
– Springer Journals

Published: Feb 6, 2009

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References

Tauberian theorems for weighted means of double sequences

Chen, C.-P.; Hsu, J.-M.

A contribution to the theory of divergent sequences

Lorentz, G.G.

Some general Tauberian theorems

Maddox, I.J.

Almost convergence of double sequences and strong regularity of summability matrices